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operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which are solutions of differential equations.
An operator can perform a function on any number of operands (inputs) though most often there is only one operand.
An operator might also be called an operation, but the point of view is different. For instance, one can say "the operation of addition" (but not the "operator of addition") when focusing on the operands and result. One says "addition operator" when focusing on the process of addition, or from the more abstract viewpoint, the function +: S×S ? S.
Notation An operator name or operator symbol is a notation which denotes a particular operator. When there is no danger of confusion, an operator name or operator symbol may be referred to more briefly as an "operator". Strictly speaking, however, the operator is a mathematical object and not the syntactic entity which denotes it. The reason for identifying it with its notation is that there are some operators which have come to have standard notations.
Unicode reserves U+2200 to U+22FF for basic "Mathematical Operators," almost all of which is defined in version 1.0 (and thus can be displayed by most internet browsers released since October 1991)
Simple examples In linear algebra an "operator" is a linear operator
. In analysis an "operator" may be a differential operator, to perform ordinary differentiation, or an integral operator, to perform ordinary integration.
One example of a differential operator is the derivative itself. The corresponding operator name D, when placed before a differentiable function f, indicates that the function is to be differentiated with respect to the variable.
Operators versus functions The word operator can in principle be applied to any function. However, in practice it is most often applied to functions which operate on mathematical entities of higher complexity than real numbers, such as vectors, random variables, or mathematical expressions. The differential and integral operators, for example, have domains and codomains whose elements are mathematical expressions of indefinite complexity. In contrast, functions with vector-valued domains but scalar ranges are called functionals and forms.
In general, if either the do
Additionally, when functions are used so often that they have evolved faster or easier notations than the generic F(x,y,z,...) form, the resulting special forms are also called operators. Examples include infix operators such as addition "+" and division "/", and postfix operators such as factorial "!". This usage is unrelated to the complexity of the entities involved.
Examples This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.
Linear operators
The most common kind of operator encountered are linear operators. In talking about linear operators, the operator is signified generally by the letters T or L. Linear operators are those which satisfy the following conditions; take the general operator T, the function acted on under the operator T, written as f(x), and the constant a:
Many operators are linear--for example, the differential operator and Laplacian operator.
Linear operators are also known as linear transformations or linear mappings. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with nonlinearity).
Such an example of a linear transformation between vectors in R2 is reflection: given a vector x = (x1, x2)
Q(x1, x2) = (−x1, x2)
We can also make sense of linear operators between generalisations of finite-dimensional vector spaces. For example, there is a large body of work dealing with linear operators on Hilbert spaces and on Banach spaces.
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